Filter apparatus

ABSTRACT

A filter designed in accordance with Laplace Transform theory and then approximated to a realizable filter having an impulse response of one-half cycle of a cosine wave.

Tliiiled States Patent [56] References Cited UNITED STATES PATENTS2,627,546 2/1953 Paine 333/20 X 2,963,647 12/1960 Dean 333/20 X OTHERREFERENCES Analysis of Linear Systems -Cheng; Addison Wesley PublishingCo., Reading, Mass. 1959; pages 166- 168 and 233 PrimaryExaminer-l-lerman Karl Saalbach Assistant Examiner-Marvin NussbaumAttorneysRobert J. Crawford and Bruce C. Lutz ABSTRACT: A filterdesigned in accordance with Laplace Transform theory and thenapproximated to a realizable filter having an impulse response ofone-half cycle of a cosine wave.

Inventor John Dan ll-lill, III

Dallas, Tex. Appl. No. 61,558 Filed Aug. 6, 1970 Patented Oct. 19, 1971Assignee Collins Radio Company Dallas, Tex.

FILTER APPARATUS 3 Claims, 2 Drawing Figs.

US. Cl 333/20, 3 3 3/70 R Int. Cl l-l03h 7/04, H03k 5/01 Field of Search333/20, 70, 73

J my L n I N P u T L L PATENTEUucT 19 an 1 3,614, 74

INVENTOR.

JOHN D. HILL ATTORNEY PATENTEUnm 19m I 3,614,674

SHEET 20F 2 FIG. ;2

INVENTO/i.

JOHN D. H/LL g/Wax;

ATTORNEY FILTER APPARATUS THE INVENTION The present invention pertainsgenerally to filters and more specifically to a filter having a sinewave response to a rectangular waveform input signal having aone-half-cycle duration. The impulse response of such a filter is aone-half-cycle cosine waveform.

In the transmission of signals over long distances it has been foundthat there is excessive intersymbol interference even when nonreturn tozero. Data bit information signals are utilized. Even more intersymbolinterference occurs when the data bits are of the return to zero type.It has accordingly been ascertained that for communications over longdistances there is less intersymbol distortion when a sine wave is usedwhich reverses in phase for a change in data significance. In otherwords a change from logic 1 to logic or vice versa. However, for actualusage of data information most computing machines are designed toutilize logic ls and logic Us in the form of data pulses or the lackthereof. Thus, there must be a conversion from the data bits to thebiphase sine wave information in the modulation section thereof. Thetransformation from biphase sine wave to square wave pulses is notparticularly difficult as circuits necessary for this transformationhave been in existence for some time. However, the transformation fromrectangular data bits to biphase sine wave information is somewhat moredifficult. To my knowledge there has been no single filter designedpreviously which will accomplish this transformation.

It is therefore an object of my invention to produce a filter which willprovide biphase sine wave information from rectangular data pulse inputsignals.

Other objects and advantages of the present invention may be ascertainedfrom a reading of the specification and appended claims in conjunctionwith the drawings wherein:

FIG. 1 is a schematic diagram of one embodiment of a filter designed inaccordance with the present invention; and

FIG. 2 is a collection of waveforms utilized in explaining theinvention.

As mentioned, FIG. 1 is a schematic diagram of a filter designed inaccordance with the present invention. An explanation in visual layterminologies is a practical impossibility. Therefore, the values of thevarious components of the embodiment of the filter shown in FIG. 1 areprovided below with no further explanation of the transition of thesignal as it passes therethrough from a one-half-cycle rectangularwaveform pulse to a full-cycle sine wave signal.

All inductances L l 95.3 nanohenries C,=l 26.65 picofarads C. 1 4.072picofarads C =S.066 icofarads C =2.584 picofarads C,= l .564 picofaradsHowever, since an impulse is defined as having infinite amplitude andzero duration such a signal is for practical purposes impossible toproduce. A much more practical signal would be a rectangular pulse. Sucha pulse might be a halfcycle pulse equal to one-half the desired sinewave. Such a waveform is shown in FIG. 2D wherein the full cycle isdefined as 1/ f The Laplace Transform of such a time waveform isEquation 1 (s) 8 Equation 2 or -s/2f) 8 Equation 3 for a negative pulse.

If we divide equation 2 into equation 1 we obtain equation 4 which maybe transformed to equation 5 Equation 5 The impulse response of a filterdesigned in accordance with the Laplace Transform of equation 5 isone-half cycle of a cosine wave. However, of more importance is the factthat such a filter will produce a sine wave in response to theapplication of a one-half-period rectangular waveform pulse.

Another way of explaining why it was decided that equation 1 should bedivided by equation 2 was to remove from the filter having the LaplaceTransform of equation 1 the portion of the filter which produced thesine wave output signals necessary to produce a square wave output.Since this portion is removed from the filter which produces a sine waveupon the application of an impulse, the application of a one-halfcyclerectangular waveform square wave will produce a single cycle of a sinewave at the output.

The filter described by the Laplace Transform of equation 5 is, however,not practically realizable. Accordingly, equation 5 must be transformedto a more workable form. By normalizing equation 5 to one radian persecond, the Laplace Transform of equation 6 is obtained.

By adding one to each side, equation 8 is obtained.

In obtaining a common denominator for the right side of the equation ofequation 8 equations 9 and 10 are obtained.

. 1r8 2 1+tanh Equation 10 may now be inverted to produce equation 11which has also been multiplied by 2 on both sides after inversion.

Equation 9 Equation 10 l-l-tanh It will now be noted from equation 12that we have a portion of original equation 6. If we now substitute theleft-hand portion of equation 1 1 into the corresponding portion ofequation 6; we obtain equation 12 Mittag-Leffler in a book titled Theoryand Problems of Complex Variables authored by Murray R. Spiegel andpublished in 1964 by Schaum Publishing Co., pages 175, 191, and 192provides the theory for expanding a hyperbolic function. Using theMittag-Lefflers expansion theorem of equa- Equation 11 Equation 12 tion13, we can express the infinite sum thereof by equation 14.

Equation 13 Equation 14 If we now let 2 in equation 14 equal 1rs/2, theargument of tank in equation 12, we obtain the following equations 15and 16; wherein equation is a simplified version of equation 15.

Equation 15 t anh 2 2 2 #232 Equation 16 We may now substitute equation16 for tanh (rs/2) in equation 12 to obtain equation 17.

Equation 17 Equation 18 If the derivative of the first term of thedenominator is taken we obtain the results of equation 19.

Equation 19 I1I[s (Zn-U co co 1 =23 1;]: [3 (27?.- 1) m We find,however, that the derivative of the first term is contained in thesecond term of the denominator and therefore equation 18 can now bewritten as equation 20.

Equation 20 By inspection of equation 19 it will be determined that uponexpansion of the term within the brackets of equation I the first termin the numerator will be s -l-l. The second term will be s +9, etc.However, the first term is of interest since this will cancel with the s+l term outside the brackets of equation 20 to eliminate the problemproducing poles on the jw axis of the S plane.

The procedures outlined in Introduction to Modern Network Synthesis byVanValkenburg published by Wiley Publishing Company and copyrighted in1960 will result in a practical realization of the filter described byequation 20. The materials in Chapters 3, 5, 8, l3, and 14 provide thedetail necessary to design a filter from the Laplace Transform. Thefilter of FIG. 1 is the result of a lO-pole approximation as provided byequation 21 by evaluating the expression for n=1 to 5.

Equation 21 Since the computations involved in solving equations 20 and21 are extensive, the normal, practical way of obtaining the filterdesign information is to program a computer to obtain the filter designparameters.

While a particular embodiment of a filter has been disclosed in thepresent application, the invention comprises the broad idea of using adesired time response, Laplace Transforms and approximation to produce apractical filter and more specifically is related to a filter which willproduce biphase data from rectangular waveform input signals.Transformation techniques are well known to those skilled in the art toproduce filters which have the same impulse response as the filter shownin FIG. 1 but have a different physical implementation. Examples wouldbe transformer or lattice networks.

I therefore wish to be limited not by the disclosure but only by thescope of the appended claims wherein:

I claim:

1. The method of designing a filter substantially in accordance withthat represented by the idealized Laplace Transform comprising the stepsof:

normalizing w to one radian per second;

transposing terms and expanding to obtain the Laplace Transformapproximating the last referenced Laplace Transform for a value of nwhich is at least equal to one to remove the poles from the jaw axis ofthe S plane; and

programming a computer to solve the second Laplace Transform. 2. Themethod of altering a half-cycle rectangular pulse to a single-cycle sinewave comprising the steps of:

passing the rectangular waveform signal through a filter constructed inaccordance with a finite number of terms of the Laplace Transform

1. The method of designing a filter substantially in accordance withthat represented by the idealized Laplace Transform comprising the stepsof: normalizing omega to one radian per second; transposing terms andexpanding to obtain the Laplace Transform approximating the lastreferenced Laplace Transform for a value of n which is at least equal toone to remove the poles from the j omega axis of the S plane; andprogramming a computer to solve the second Laplace Transform.
 2. Themethod of altering a half-cycle rectangular pulse to a single-cycle sinewave comprising the steps of: passing the rectangular waveform signalthrough a filter constructed in accordance with a finite number of termsof the Laplace Transform
 3. Apparatus of the class described comprisingin combination: means for supplying an input signal having a waveformwhich is rectangular and one-half cycle of a given frequency induration; output means; and filter means connected between said inputmeans and said output means for altering said rectangular waVeform inputsignals to a single cycle of a sine wave, said filter means beingconstructed in accordance with the Laplace Transform: